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SOME MISCONCEPTIONS ABOUT ENTROPY

• Introduction — the ground rules.

• Gibbs versus Boltmann entropies.

• That awful H-theorem.

• The Second Law of Thermodynamics.

• Tsallis and other heresies.

• Open the quantum box.

• Time asymmetry and the approach to equilibrium.

• Conclusions.

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THERMODYNAMICS AND STATISTICAL MECHANICS

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STATISTICAL MECHANICS — GIBBS VS. BOLTZMANN

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INFERENCE — A BAYESIAN PERSPECTIVE

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INFERENCE AND STATISTICAL MECHANICS

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GIBBS VS. BOLTZMANN ENTROPIES

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GIBBS VS. BOLTZMANN ENTROPIES II

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THAT AWFUL H-THEOREM IN ALL THE BOOKS

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THE SECOND LAW OF THERMODYNAMICS

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PROOF OF THE SECOND LAW OF THERMODYNAMICS

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THE THEORETICAL SECOND LAW

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ENTROPY INCREASE AND THE ARROW OF TIME

• Boltzmann’s constant (kB =1.38×10−23 J K−1) is

rather small, and the increase of phase-space volume is

usually very large.

• (Taken from a Cambridge Part IB Physics Example

Sheet.) Suppose that an infrared photon of energy 1 eV is

absorbed by a dust grain at 300 K. The entropy increases

by 5.34×10−22 J K−1.

• This increases the available phase-space volume of the

Universe by a factor of exp(38.7)=6.3×1016.

• Scaled up to the size of this lecture theatre, we get a

staggering exp(1021) increase in volume PER SECOND.

• Macroscopic irreversibility is therefore not surprising in the

least. . .

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TSALLIS ENTROPIES AND OTHER HERESIES

• There has been a lot of papers published concerning the

Tsallis generalised entropy.

• These are one-parameter family of functions based on the

q-derivative, and include the correct entropy as a special

case q =1.

• The “entropic” functions for other values of the parameter q

are non-extensive (i.e. do not satisfy the Kangaroo axiom).

• The maximised “entropies” for q 6=1 are not equal to the

experimental entropy defined by Clausius.

• The industry of q-generalisations has taken hold in many

different fields. It is mathematically self-consistent, and

may be great fun, but its track record for concrete

achievements is (in my opinion) still zero.

• That said, Constantino Tsallis has published a very

impressive list of applications throughout physics and

astrophysics.

• But I don’t believe him. . .

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OPEN THE QUANTUM BOX

• Quantum particle in ground state of 1-dimensional box

− 1

2a<x< 1

2a. At t=0 the box is opened to

−a<x<a.

• We can expand the old ground state in terms of the new

states. It is still in a pure state, though not an energy

eigenstate, and the entropy is still zero.

• Evolution of box wavefunction after a long time. The

probability either has a central hump, or two symmetrical

ones, as the wavefunction displays the interference of the

2 lowest frequency modes.

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OPEN THE QUANTUM BOX II

• A movie shows that the particle oscillates around enjoying

its new-found freedom. It doesn’t remotely settle down.

Some authors average the phases and thereby get an

entropy increase of 0.683714.

• The time average (500 samples) of the probability (red

line) is almost indistinguishable from the phase average.

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OPEN THE QUANTUM BOX III

• If the box is opened one-sidedly the oscillations are more

violent. The averaged distribution has two humps and the

phase-averaged entropy increase is 1.03500.

• But this is NOT thermodynamics.

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OPEN THE QUANTUM BOX IV

• The canonical distribution with the same average energy

has an entropy of 1.594.

• If we impose symmetry on the wavefunction the entropy

increase is 1.0414 and there is a hump in the middle.

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TIME ASYMMETRY IN PHYSICS

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TIME ASYMMETRY AND NON-EQUILIBRIA

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BROWNIAN MOTION

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EQUILIBRIUM ENSEMBLE FOR BROWNIAN MOTION

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BROWNIAN MOTION AND TIME ASYMMETRY

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UNCERTAINTY VERSUS FLUCTUATIONS

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CONCLUSIONS

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APPENDIX — PROOF OF SG =SE

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APPENDIX — PROOF OF SG =SE II

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BROWNIAN MOTION